Logic, Non-Classical
The purpose of this entry is to survey those modern logics that are often called "non-classical," classical logic being the theory of validity concerning truth functions and first-order quantifiers likely to be found in introductory textbooks of formal logic at the end of the twentieth century.
For the sake of uniformity I will give a model-theoretic account of the logics. All of the logics also have proof-theoretic characterizations, and in some cases (such as linear logic) these characterizations are somewhat more natural. I will not discuss combinatory logic, which is not so much a non-classical logic as it is a way of expressing inferences that may be deployed for both classical and non-classical logics. I will use A, B, … for arbitrary sentences; ∧, ∨, ¬, and →, for the standard conjunction, disjunction, negation, and conditional operators for whichever logic is at issue. "Iff" means "if and only if." For references see the last section of this article.
Extensions Versus Rivals
An important distinction is that between those non-classical logics that take classical logic to be alright as far as it goes, but to need extension by the addition of new connectives, and those which take classical logic to be incorrect, even for the connectives it employs.
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